Recall the transformation
\(\widetilde{T}(\widetilde{x})=(t,\sigma _t(\mathbf {v}),\kappa _{t,\mathbf {v}}(\mathbf {\omega })), \ \widetilde{x}=(t,\mathbf {v},\mathbf {\omega })\in \widetilde{X}.\) 
Then we have for \(k\ge 1\) ,
\(M\circ \widetilde{T}^k(\widetilde{x})=&\ S_{R^k(t)}\Phi (\bar{\theta }^+_{R^k(t),\sigma _{k,t}\mathbf {v},\mathbf {\omega }_{\mathbf {v}^+|_{k,t}}}).\) 

Let \(\mathcal {M}\)  be the set of all probability measures on the closed ball \(B(0,R_E)\)  in \(\mathbb {R}^2\) , where we denote by \(R_E=|E|\) . For any bounded continuous function \(f\)  on \(\mathcal {M}\)  (w.r.t the weak-topology), by Lemma REF  and Birkhoff's ergodic theorem we have for \(\mathbb {Q}_\theta \) -a.e. \(\widetilde{x}\in \widetilde{X}\) ,
\(\lim _{n\rightarrow \infty }\frac{1}{n}\sum _{k=0}^{n-1}f(M\circ \widetilde{T}^k(\widetilde{x}))=\mathbb {E}_{\mathbb {Q}_\theta }(f(M)).\) 
In fact, since the space of continuous functions on \(\mathcal {M}\)  is separable under \(\Vert \cdot \Vert _\infty \)  norm, the above limit holds for all continuous functions on \(\mathcal {M}\)  simultaneously (see [1]} for example).
Fix \(q\ge 1\) . Due to the lower semicontinuity of \(H_{\rho ^q}(\pi _s(\cdot ))\) , we may find a countable dense sequence of continuous functions \(f_k\)  on \(\mathcal {M}\)  that can approximate \(H_{\rho ^q}(\pi _s (\cdot ))\)  from below for all \(s\in \mathbb {R}\) . Then, by (REF ) we have for \(\mathbb {Q}_\theta \) -a.e. \(\widetilde{x}\in \widetilde{X}\) , for all \(s\in \mathbb {R}\) ,
\(\liminf _{n\rightarrow \infty }\frac{1}{n}\sum _{k=0}^{n-1}\frac{1}{\log (1/\rho )}H_{\rho ^q}(\pi _s M\circ \widetilde{T}^k(\widetilde{x}))\ge qE_{q,\theta }(\pi _s).\) 
Considering the subsequence \(qn\rightarrow \infty \) , we may rewrite the above summation as
\(\frac{1}{qn}\sum _{k=0}^{qn-1}\frac{1}{\log (1/\rho )}H_{\rho ^q}(\pi _s M\circ \widetilde{T}^k(\widetilde{x}))=\sum _{l=0}^{q-1} \frac{1}{n}\sum _{k=0}^{n-1}\frac{1}{q}\frac{1}{\log (1/\rho )}H_{\rho ^q}(\pi _s M\circ \widetilde{T}^{l+qk}(\widetilde{x})).\) 
Then by (REF ) we have that for \(\mathbb {Q}_\theta \) -a.e. \(\widetilde{x}\in \widetilde{X}\) , for all \(s\in \mathbb {R}\) , there exists at least one \(l\in \lbrace 0,\ldots ,q-1\rbrace \)  such that
\(\liminf _{n\rightarrow \infty }\frac{1}{n}\sum _{k=0}^{n-1}\frac{1}{q\log (1/\rho )}H_{\rho ^q}(\pi _s M\circ \widetilde{T}^{l+qk}(\widetilde{x})) \ge E_{q,\theta }(\pi _s) .\) 
We shall use the \(\rho \) -tree method in [2]} to obtain lower bounds for the dimensions of projections of measures. Fix an instance of \(\widetilde{x}=(t,\mathbf {v},\mathbf {\omega })\in \widetilde{X}\) , \(s\in \mathbb {R}\)  and \(l\in \lbrace 0,\ldots ,q-1\rbrace \)  such that (REF ) holds. Shortly denote by
\(\bar{\theta }^{\prime }=\bar{\theta }^+_{R^l(t),\sigma _{l,t}\mathbf {v},\mathbf {\omega }_{\mathbf {v}^+|_{l,t}}}, \mathbf {w}=\sigma _{l,t}(\mathbf {v}^+), t^{\prime }=R^l(t).\) 
For \(k\ge 1\)  consider finite partition of \(\Sigma _{a,b}^+\) :
\(\mathcal {P}_{t^{\prime }}^{qk}=\lbrace [\mathbf {v}^+|_{qk,t^{\prime }}]=[\mathbf {i}^+|_{\tau _{qk}(t^{\prime })}]\times [\mathbf {j}^+|_{qk}]: \mathbf {v}^+=(\mathbf {i}^+,\mathbf {j}^+)\in \Sigma _{a,b}^+\rbrace .\) 
We may map \(\mathcal {P}_{t^{\prime }}^{qk}\)  to \(\mathbb {N}^k\)  for each \(k\ge 1\)  in a way such that each element of \(\mathcal {P}_{t^{\prime }}^{qk}\)  is represented by a word of length \(k\) , and for \(u\in \mathbb {N}^k\)  that represents an element \(C_u\in \mathcal {P}_{t^{\prime }}^{qk}\) , and for \(u^{\prime }\in \mathbb {N}^{k+1}\)  that represents an element \(C_{u^{\prime }}\in \mathcal {P}_{t^{\prime }}^{q(k+1)}\) , if \(C_{u^{\prime }}\subset C_{u}\) , then \(u^{\prime }|_k=u\) . All the words that represent \(\lbrace \mathcal {P}_{t^{\prime }}^{qk}\rbrace _{k\ge 1}\)  then form the set of finite words of a tree in \(\mathbb {N}^\mathbb {N}\)  denoted by \(Y_{q,t^{\prime }}\) . We equip \(Y_{q,t^{\prime }}\)  with the metric \(d_{\rho ^q}\)  so that it becomes a \(\rho ^q\) -tree. Note also that for each \(\mathbf {v}^+\in \Sigma _{a,b}^+\)  there is a unique sequence \(\lbrace [\mathbf {v}^+|_{qk,t^{\prime }}]\rbrace _{k\ge 1}\)  in \(\lbrace \mathcal {P}_{t^{\prime }}^{qk}\rbrace _{k\ge 1}\)  that converges to \(\lbrace \mathbf {v}^+\rbrace \) , therefore there is a one-to-one correspondence between \(Y_{q,t^{\prime }}\)  and \(\Sigma _{a,b}^+\) . Hence we may identify \(Y_{q,t^{\prime }}\)  with \(\Sigma _{a,b}\) , and for \(k\ge 1\) , \([\mathbf {v}^+|_{qk,t^{\prime }}]\)  can be viewed as the cylinder of generation \(k\)  in the \(\rho ^q\) -tree \(Y_{q,t^{\prime }}\)  that contains \(\mathbf {v}^+\) .
For a measure \(\lambda \)  on \(Y_{q,t^{\prime }}\)  and a finite word \(u\)  in \(Y_{q,t^{\prime }}\)  with \(\lambda ([u])>0\)  we denote by
\(\lambda _{[u]}(B)=\frac{\lambda ([u]\cap B)}{\lambda ([u])},\ B\in \mathcal {B}_{Y_{q,t^{\prime }}}\) 
the measure \(\lambda \)  conditioned on \([u]\) , and
\(\lambda ^{[u]}(B)=\frac{\lambda (uB)}{\lambda ([u])}, \ B\in \mathcal {B}_{Y_{q,t^{\prime }}}.\) 
the zooming-in of \(\lambda \)  at \(u\) , where \(uB=\lbrace uy:y\in B\rbrace \) . Then by (REF ) we have
\(M\circ \widetilde{T}^{l+qk}(\widetilde{x})=&\ S_{R^{qk}(t^{\prime })}\Phi (\bar{\theta }^+_{t,\mathbf {v}^-\mathbf {v}^+|_{l,t}\mathbf {w}^+|_{t^{\prime },qk},\mathbf {\omega }_{\mathbf {v}^+|_{l,t}\mathbf {w}|_{t^{\prime },qk}}})\\=&\ S_{\varphi ^{qk}(t^{\prime })}\Phi ((\overline{\theta }^{\prime })^{[\mathbf {w}|_{qk,t^{\prime }}]}).\) 

Hence (REF ) implies for \(\bar{\theta }^{\prime }\) -a.e. \(\mathbf {w}\in Y_{q,t^{\prime }}\) ,
\(\liminf _{n\rightarrow \infty }\frac{1}{n}\sum _{k=0}^{n-1}\frac{1}{q\log (1/\rho )}H_{\rho ^q}(\pi _s S_{\varphi ^{qk}(t^{\prime })}\Phi ((\bar{\theta }^{\prime })^{[\mathbf {w}|_{qk,t^{\prime }}]})) \ge E_{q,\theta }(\pi _s) .\) 
By (REF ), the canonical map \(\Phi \colon (Y_{q,t^{\prime }},d_{\rho ^q})\rightarrow E\)  is \(c_E\) -Lipschitz, where \(c_E=R_E^2c_\delta \)  for \(c_\delta :=\sqrt{\delta ^2+1}\) . Indeed, let \(\mathbf {v}^1=(\mathbf {i}^1,\mathbf {j}^1),\mathbf {v}^2=(\mathbf {i}^2,\mathbf {j}^2)\in Y_{q,t^{\prime }}\)  be such that \(d_{\rho ^q}(\mathbf {v}^1, \mathbf {v}^2)=\rho ^{qk}\) . This implies that their first \(k\)  letters must coincide, i.e., \(\mathbf {v}^1|_{qk,t^{\prime }}=\mathbf {v}^2|_{qk,t^{\prime }}\) . This means that \(f_{\mathbf {i}^1|_{qk,t^{\prime }}}=f_{\mathbf {i}^2|_{qk,t^{\prime }}}\)  and \(g_{\mathbf {j}^1|_{qk}}=g_{\mathbf {j}^2|_{qk}}\)  for \(l=1,2\) , and in particular they form the same translation on \(\mathbb {R}^2\) . Then,
\(|\Phi (\mathbf {v}^1)-\Phi (\mathbf {v}^2)|&\le |\Phi ([\mathbf {v}^1|_{qk,t^{\prime }}])|\cdot |\Phi \circ \sigma _{qk,t^{\prime }}(\mathbf {v}^1)-\Phi \circ \sigma _{qk,t^{\prime }}(\mathbf {v}^2)|\\&\le R_Ec_\delta \rho ^{qk} \cdot R_E\\&=R_E^2c_\delta \cdot d_{\rho ^q}(\mathbf {v}^1,\mathbf {v}^2).\) 

For \(k\ge 1\)  denote by
\(A^{qk}_{t^{\prime }}=\begin{pmatrix}\delta ^{\tau _{qk}(t^{\prime })} & 0 \\0 & \rho ^{qk} \\\end{pmatrix}.\) 
Then using (REF ) we have for \(\mathbf {v}^1,\mathbf {v}^2\in [\mathbf {w}|_{qk,t^{\prime }}]\) ,
\(|\pi _sS_{R^{qk}(t^{\prime })}\Phi \circ \sigma _{qk,t^{\prime }}(\mathbf {v}^1)-\pi _sS_{R^{qk}(t^{\prime })}\Phi \circ \sigma _{qk,t^{\prime }}&(\mathbf {v}^2)|\\&=|\pi _sS_{t^{\prime }}(A^{qk}_{t^{\prime }})^{-1}\Phi (\mathbf {v}^1)-\pi _sS_{t^{\prime }}(A^{qk}_{t^{\prime }})^{-1}\Phi (\mathbf {v}^2)|\\&\le c_\delta \rho ^{-qk}|\pi _sS_{t^{\prime }}\Phi (\mathbf {v}^1)-\pi _sS_{t^{\prime }}\Phi (\mathbf {v}^2)|.\) 

This tells us that for \(\mathbf {v}^1,\mathbf {v}^2\in [\mathbf {w}|_{qk,t^{\prime }}]\) , if
\(|\pi _sS_{t^{\prime }}\Phi (\mathbf {v}^1)-\pi _sS_{t^{\prime }}\Phi (\mathbf {v}^2)|\le c_\delta \rho ^{q(k+1)},\) 
then
\(|\pi _sS_{R^{qk}(t^{\prime })}\Phi \circ \sigma _{qk,t^{\prime }}(\mathbf {v}^1)-\pi _sS_{R^{qk}(t^{\prime })}\Phi \circ \sigma _{qk,t^{\prime }}(\mathbf {v}^2)|\le \rho ^q.\) 
Therefore for \(\mathbf {v}^1\in [\mathbf {w}|_{qk,t^{\prime }}]\) ,
\(\lbrace \mathbf {v}^2\in & [\mathbf {w}|_{qk,t^{\prime }}]:|\pi _sS_{t^{\prime }}\Phi (\mathbf {v}^1)-\pi _sS_{t^{\prime }}\Phi (\mathbf {v}^2)|\le c_\delta \rho ^{q(k+1)}\rbrace \\&\subset \lbrace \mathbf {v}^2 \in [\mathbf {w}|_{qk,t^{\prime }}]:|\pi _sS_{R^{qk}(t^{\prime })}\Phi \circ \sigma _{qk,t^{\prime }}(\mathbf {v}^1)-\pi _sS_{R^{qk}(t^{\prime })}\Phi \circ \sigma _{qk,t^{\prime }}(\mathbf {v}^2)|\le \rho ^q\rbrace .\) 

This implies that
\(\nonumber H_{\rho ^q}(\pi _s S_{R^{qk}(t^{\prime })}\Phi ((\bar{\theta }^{\prime })^{[\mathbf {w}|_{qk,t^{\prime }}]}))=&\ H_{\rho ^q}(\pi _sS_{R^{qk}(t^{\prime })}(\Phi \circ \sigma _{qk,t^{\prime }})(\bar{\theta }^{\prime })_{[\mathbf {w}|_{qk,t^{\prime }}]}))\\\le &\ H_{c_\delta \rho ^{q(k+1)}}(\pi _sS_{t^{\prime }}\Phi ((\overline{\theta }^{\prime })_{[\mathbf {w}|_{qk,t^{\prime }}]})), \) 

where we have used the fact that, by definition,
\((\bar{\theta }^{\prime })^{[\mathbf {w}|_{t^{\prime },qk}]}=(\bar{\theta }^{\prime }\circ \sigma _{qk,t^{\prime }}^{-1})_{[\mathbf {w}|_{qk,t^{\prime }}]}.\) 
At last we derive that for \(\bar{\theta }^{\prime }\) -a.e. \(\mathbf {w}\in Y_{q,t^{\prime }}\) ,
\(\liminf _{n\rightarrow \infty }\frac{1}{n}\sum _{k=0}^{n-1}\frac{1}{q\log (1/\rho )}H_{c_\delta \rho ^{q(k+1)}}(\pi _sS_{t^{\prime }}\Phi ((\overline{\theta }^{\prime })_{[\mathbf {w}|_{qk,t^{\prime }}]})) \ge E_{q,\theta }(\pi _s). \) 
The mapping \(f_{q,t^{\prime },s}:=\pi _sS_{t^{\prime }}\Phi \colon (Y_{q,t^{\prime }},d_{\rho ^q})\rightarrow \mathbb {R}\)  is \(c_E\) -Lipschitz. By [2]}, there exists a \(\rho ^q\) -tree \((Y_{q,t^{\prime },s},d_{\rho ^q})\)  and maps \(Y_{q,t^{\prime }}\xrightarrow{}Y_{q,t^{\prime },s}\xrightarrow{}\mathbb {R}\)  such that \(f_{q,t^{\prime },s}=f^{\prime }_{q,t^{\prime },s}h_{q,t^{\prime },s}\) , where \(h_{q,t^{\prime },s}\)  is a tree morphism and \(f^{\prime }_{q,t^{\prime },s}\)  is \(C\) -faithful (see [2]}) for some constant \(C\)  depending only on \(c_E\) . Then applying [2]} to the \(\rho ^q\) -tree \((Y_{q,t^{\prime },s},d_{\rho ^q})\)  (for which \(f^{\prime }_{q,t^{\prime },s}\)  is \(C\) -faithful), and [2]}, there is a constant \(C^{\prime }\)  depending only on \(C\)  and \(c_\delta \) , such that for all \(k\ge 1\) ,
\(|H_{c_\delta \rho ^{q(k+1)}}(f_{q,t^{\prime },s}((\bar{\theta }^{\prime })_{[\mathbf {w}|_{qk,t^{\prime }}]})-H_{\rho ^{q(k+1)}}(h_{q,t^{\prime },s}((\bar{\theta }^{\prime })_{[\mathbf {w}|_{qk,t^{\prime }}]})|\le C^{\prime }.\) 
Consequently, for \(\bar{\theta }^{\prime }\) -a.e. \(\mathbf {w}\in Y_{q,t^{\prime }}\) ,
\(\frac{1}{q\log (1/\rho )}\liminf _{n\rightarrow \infty }\frac{1}{n}\sum _{k=0}^{n-1}H_{\rho ^{q(k+1)}}(h_{q,t^{\prime },s}((\bar{\theta }^{\prime })_{[\mathbf {w}|_{qk,t^{\prime }}]})\ge E_{q,\theta }(\pi _s)-O(1/q),\) 

where the constant in \(O(1/q)\)  only depends on \(\rho \)  and \(C^{\prime }\) . Then, by [2]}, it follows that
\(\underline{\dim }_H h_{q,t^{\prime },s}(\bar{\theta }^{\prime }) \ge E_{q,\theta }(\pi _s)-O(1/q).\) 
Since \(f^{\prime }_{q,t^{\prime },s}\)  is \(C\) -faithful and \(f^{\prime }_{q,t^{\prime },s}h_{q,t^{\prime },s}=f_{q,s,t^{\prime }}=\pi _sS_{t^{\prime }}\Phi \) , we get from [2]} that
\(\underline{\dim }_H \pi _sS_{t^{\prime }}\Phi (\bar{\theta }^{\prime })\ge E_{q,\theta }(\pi _s)-O(1/q).\) 
Finally, as
\(S_{t^{\prime }}\Phi (\bar{\theta }^{\prime })=S_{t^{\prime }}\Phi (\bar{\theta }^+_{R^l(t),\sigma _{l,t}\mathbf {v},\mathbf {\omega }_{\mathbf {v}^+|_{l,t}}})=S_{t}\Phi ((\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }})^{[\mathbf {v}^+|_{l,t}]}),\) 
we deduce that for \(\mathbb {Q}_\theta ^-\) -a.e. \((t,\mathbf {v}^-,\mathbf {\omega })\in \mathbb {T}\times \Sigma _{a,b}^-\times \Omega _{a,b}\) , for all \(s\in \mathbb {R}\)  and \(q\ge 1\) ,
\(\underline{\dim }_H \pi _sS_{t}\Phi (\bar{\theta }^+_{t,\mathbf {v}^-,\mathbf {\omega }})\ge E_{q,\theta }(\pi _s)-O(1/q).\) 
